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CH 31 SQUARES AND CIRCLES

Review of Formulas

If s is the length of each side of a square,

then its perimeter is 4s and its area is s2.

If r is the radius of a circle, then its

diameter is 2r, its circumference is 2r, and

its area is r2.

Homework

1. True/False: Every square is a rectangle.

2. True/False: Every rectangle is a square.

3. Each side of a square is 17. Find the square’s perimeter and

area.

4. The radius of a circle is 10. Find the diameter, the circumference,

and the area in exact form (that means leave as ).

5. Each side of a square is 25. Find the square’s perimeter and

area.

6. The radius of a circle is 1. Find the diameter, the circumference,

and the area.

7. The radius of a circle is 1

2. Find the diameter, the circumference,

and the area.

r

s

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Squares

EXAMPLE 1: The perimeter of a square is 94. What is the

length of each side of the square?

Solution: The formula for the perimeter of a square is

P = 4s

Substituting the given information, we get

94 = 4s

Dividing each side of the equation by 4 gives

94 4=4 4

s , or

EXAMPLE 2: The area of a square is 361. How long is each

side of the square?

Solution: The area of a square is given by A = s2. Plugging 361

in for A yields the equation

361 = s2

We need to take the square root of 361 in order to solve for s.

Using a calculator (or clever guessing), we conclude that

s = 23.5

s = 19

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EXAMPLE 3: The perimeter of a square is 50.4. Find the area

of the square.

Solution: We’re being asked for the area of a square. But the

area formula, A = s2, requires that we know the side, and we

don’t. However, we are given the perimeter. Do you see what

we’re going to do? We will use the perimeter to find the side, and

then use the side to find the area.

P = 4s (perimeter of a square)

50.4 = 4s (put in the given area)

s = 12.6 (divide each side by 4)

Therefore,

A = s2 (area of a square)

= (12.6)2 (put in the s just calculated)

= 158.76 (multiply 12.6 by itself)

We conclude that the area is

EXAMPLE 4: The area of a square is 625. What is the

square’s perimeter?

Solution: This is the reverse of the previous example, but we

solve it in essentially the same way: We’ll use the given area to

determine the side, and then use the side to find the perimeter.

A = s2 625 = s2 s = 25

And so,

P = 4s = 4(25) =

158.76

100

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Homework

8. For each problem, the perimeter of a square is given. Find

the side of the square. You may use a calculator.

a. P = 144 b. P = 49.2 c. P = 210 d. P = 0.092

e. P = 92 f. P = 10.4 g. P = 504 h. P = 0.04

9. For each problem, the area of a square is given. Find the

side of the square. You may use a calculator.

a. A = 144 b. A = 6.76 c. A = 0.0625 d. A = 6,241

e. A = 529 f. A = 6.25 g. A = 0.0016 h. A = 1,764

10. For each problem, the perimeter of a square is given. Find

the area of the square. You may use a calculator.

a. P = 48 b. P = 50 c. P = 9.6 d. P = 201.2

e. P = 100 f. P = 30 g. P = 20.8 h. P = 165.2

11. For each problem, the area of a square is given. Find the

perimeter of the square. You may use a calculator.

a. A = 121 b. A = 2116 c. A = 18.49 d. A = 0.09

e. A = 196 f. A = 841 g. A = 12.25 h. A = 0.0025

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Circles

Deriving the Circumference Formula: C = 2r

The next problem we’re going to solve using our algebra skills is to

derive the formula C = 2r for the circumference of a circle with radius r.

We begin with the definition of from earlier in the class, namely that

is the ratio of the circumference of any circle to its diameter:

= Cd

Multiplying each side of this equation by d gives:

=[ ] C

dd [d ]

Simplifying gives

d = C

Now we turn the equation around and also change d into 2r, since the

diameter is twice the radius:

C = (2r)

Applying the fact that multiplication is an associative operation, we can

“shift” the parentheses to associate the and the 2:

C = (2)r

Since multiplication is also a commutative operation, we can switch the

and the 2 to get:

C = (2)r

Again, since multiplication is an associative operation, the parentheses

are redundant (unnecessary) and we can remove them. Our final

formula:

C = 2r

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given in the problem

EXAMPLE 5: The diameter of a circle is 12.

Find the area.

Solution: The formula for the area of a

circle, A = r2, requires that we have the

radius, which was not explicitly given to us in

the problem. But the diameter of the circle

was given, so we can use this diameter to find

the radius, and then use the radius to

find the area:

12= = =2 2dr 6

Now we know that r = 6, so the area is calculated as follows:

2 2= = 6A r =

EXAMPLE 6: Find the circumference of a circle whose

diameter is 9.

Solution: The process is analogous to the previous problem.

= =2dr 9

2, and so

9= 2 = 2 = 22

C r

92

=

EXAMPLE 7: The circumference of a circle is 24. Find the

radius.

Solution: The circumference of a circle is given by C = 2r.

Also, the circumference is given to be 24. So if we set the

36

9

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circumference formula (2r) equal to the given circumference

(24), we should be able to solve for r:

2 = 24r (C = 2r)

2 24=2 2

r

(divide each side by 2)

2

2

r

24=

21

12

(divide out common factors)

(and we’ve isolated the r)

EXAMPLE 8: Find the radius of a circle if it’s known that its

area is 121.

Solution: We set the area formula (r2) equal to the given area

(121), and solving for r will be as easy as .

2 = 121r

2r

121=

2 = 121r

Using a calculator, or better yet, your brain, we see that

Homework

12. Find the circumference and the area of the circle whose

diameter is given:

a. d = 10 b. d = 13 c. d = 2 d. d = 200

13. Find the radius of the circle given its circumference or area:

a. C = 30 b. C = 19 c. A = 100 d. A =

r = 12

r = 11

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EXAMPLE 9: The circumference of a circle is 4.6. Find the

area.

Solution: First ask yourself, What do I need in order to find the

area? The radius, of course. But was the radius given to us? No,

but the circumference was, so we can use the circumference to

find the radius, and then use that radius to find the area.

Step 1: Use the circumference to find the radius The circumference is given to be 4.6.

2r = 4.6 (circumference = 2r)

2 4.6=2 2

r

(divide each side by 2)

r = 2.3 (simplify each side)

Step 2: Use the radius to find the area

2 2= = (2.3) =A r

EXAMPLE 10: The area of a circle is 2.25.

Find the circumference.

Solution: Reversing the logic in the previous problem, we use

the given area to calculate the radius, and then use that radius to

find the circumference.

2

2 22.25= 2.25 = = 2.25 =rr r r

1.5

It thus follows that = 2 = 2 (1.5) = 2(1.5) =C r

5.29

3

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Homework

14. Find the area of the circle with the given circumference:

a. C = 10 b. C = 2 c. C = d. C = 5

15. Find the circumference of the circle with the given area:

a. A = 25 b. A = c. A = 81 d. A = 225

16. a. Find the circumference of a circle with diameter 2.

b. Find the area of a circle with diameter 30.

c. Find the radius of a circle with circumference 38.

d. Find the diameter of a circle with circumference 8.

e. Find the radius of a circle with area 576.

f. Find the diameter of a circle with area 400.

g. Find the area of a circle with circumference 2.

h. Find the circumference of a circle with area 16.

i. Find the circumference of a circle with diameter 28.

j. Find the area of a circle with diameter 28.

k. Find the radius of a circle with circumference 34.

l. Find the diameter of a circle with circumference 20.

m. Find the radius of a circle with area 529.

n. Find the diameter of a circle with area 81.

o. Find the area of a circle with circumference 18.

p. Find the circumference of a circle with area 169.

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Review Problems

17. The perimeter of a square is 73.6. Find the side of the square.

18. The area of a square is 10,000. Find the side of the square.

19. Find the area of a square whose perimeter is 180.

20. Find the perimeter of a square whose area is 6.25.

21. The circumference of a circle is 20. Find the area.

22. The area of a circle is 49. Find the circumference.

23. True/False:

a. If the area of a square is 25, its perimeter is 20.

b. If the perimeter of a square is 16, its area is 16.

c. If the diameter of a circle is 12, its area is 144.

d. If the diameter of a circle is 20, its circumference is 20.

e. If the circumference of a circle is 20, its area is 100.

f. If the area of a circle is 49, its diameter is 14.

g. If the radius of a circle is 11, its area is 121.

h. If the area of a circle is 36, its circumference is 36.

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Solutions

1. True 2. False 3. P = 68; A = 289

4. d = 20; C = 20; A = 100 5. P = 100; A = 625

6. d = 2; C = 2; A = 7. d = 1; C = ; 14

=A

8. a. s = 36 b. s = 12.3 c. s = 52.5 d. s = 0.023

e. s = 23 f. s = 2.6 g. s = 126 h. s = 0.01

9. a. s = 12 b. s = 2.6 c. s = 0.25 d. s = 79

e. s = 23 f. s = 2.5 g. s = 0.04 h. s = 42

10. a. A = 144 b. A = 156.25 c. A = 5.76 d. A = 2530.09

e. A = 625 f. A = 56.25 g. A = 27.04 h. A = 1705.69

11. a. P = 44 b. P = 184 c. P = 17.2 d. P = 1.2

e. P = 56 f. P = 116 g. P = 14 h. P = 0.2

12. a. C = 10; A = 25 b. C = 13; A = 42.25

c. C = 2; A = d. C = 200; A = 10,000

13. a. r = 15 b. r = 9.5 c. r = 10 d. r = 1

14. a. A = 25 b. A = c. A = 0.25 d. A = 6.25

15. a. C = 10 b. C = 2 c. C = 18 d. C = 30

16. a. C = 2 b. A = 225 c. r = 19 d. d = 8

e. r = 24 f. d = 40 g. A = h. C = 8

i. C = 28 j. A = 196 k. r = 17 l. d = 20

m. r = 23 n. d = 18 o. A = 81 p. C = 26

17. 18.4 18. 100 19. 2025

20. 10 21. 100 22. 14

23. a. T b. T c. F d. T

e. T f. T g. F h. F

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To and Beyond!

A. Find a square whose area is equal to its perimeter (ignoring the fact

that units of area are different from the units of perimeter).

B. Find a rectangle (which is not a square) whose area is equal to its

perimeter (ignoring the fact that units of area are different from units

of perimeter).

C. Find a circle whose area is equal to its circumference (even though the

units must be different).

D. Find the radius of a circle whose circumference is 20.

E. Find the radius of a circle whose area is 10.

F. By what factor must the radius of a circle be increased in order to

increase the circumference by a factor of 4?

G. By what factor must the radius of a circle be increased in order to

increase the area by a factor of 9?

“The wisest mind

has something

yet to learn.”

George Santayana (1863 - 1952)